Mean First Passage Times for Transport Equations

Abstract

Many transport processes in ecology, physics and biochemistry can be described by the average time to first find a site or exit a region, starting from an initial position. Typical mathematical treatments are based on formulations that allow for various diffusive forms and geometries but where only initial and final positions are taken into account. Here, we develop a general theory for the mean first passage time (MFPT) for velocity jump processes. For random walkers, both position and velocity are tracked and the resulting Fokker-Planck equation takes the form of a kinetic transport equation. Starting from the forward and backward formulations we derive a general elliptic integro-PDE for the MFPT of a random walker starting at a given location with a given velocity. We focus on two scenarios that are relevant to biological modelling; the diffusive case and the anisotropic case. For the anisotropic case we also perform a parabolic scaling, leading to a well known anisotropic MFPT equation. To illustrate the results we consider a two-dimensional circular domain under radial symmetry, where the MFPT equations can be solved explicitly. Furthermore, we consider the MFPT of a random walker in an ecological habitat that is perturbed by linear features, such as wolf movement in a forest habitat that is crossed by seismic lines.